Electron Diffraction
Do moving electrons display wave nature?
To answer this question you will direct a beam of electrons through a thin layer of
carbon and analyze the resulting pattern.
Theory
Louis de Broglie originated the idea that moving electrons may exhibit both
particle and wave nature. He proposed that, analogous to photons, the wavelength of
the electron is given by:
h
h
λ= =
p mv
(1)
where h is Planck's constant and p = mv is the nonrelativistic electron momentum.
This experiment is an attempt to see whether electrons really do act like waves or
whether they always act like particles.
θ
If electrons behave as particles (rigid spheres), the distribution of electrons will vary
continuously as a function of angle as in Figure 1. This distribution will vary only
slightly with changes in electron energy.
carbon target
I(θ)
electron beam
Figure 1. Particle Model Diffraction.
Continuous distribution of electrons as a function of angle.
If electrons behave as a wave, however, a diffraction pattern will emerge. We can make
an analogy with the diffraction of x-rays by a crystal.
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Electron Diffraction
Review of x-ray diffraction
Crystals act as three-dimensional gratings; they scatter the wave and produce
observable interference effects. In fact, x-rays scatter from the lattice planes (or Bragg
planes) of the crystal as shown in Figure 2.
Figure 2. Electron Waves Reflected from Atomic Planes.
There are two conditions for constructive interference of waves:
1.
The angle of incidence must equal the angle of reflection
2.
The difference in path length must be an integral number of wavelengths
nλ = 2 d sin θ.
Bragg’s Law
(2)
where n is an integer and is the order of diffraction.
By looking at the diffraction pattern and calculating the angle of diffraction, Bragg’s
Law allows us to determine either:
•
the wavelength of the x-rays if the crystal structure is known, or
•
the interatomic spacing of the crystal if the x-ray wavelength is known.
The second method is often used to determine the structure of an unknown crystal by
performing an x-ray diffraction experiment.
If electrons act like waves, we should be able to apply Bragg’s Law to the diffraction of
electrons. In that case the beam would appear as concentric rings around a bright
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Electron Diffraction
center. This pattern can be thought of as a one-dimensional diffraction pattern of bright
spots rotated about its center.
θ
carbon target
electron beam
θ)I(
Figure 3. Wave Model Diffraction.
Constructive interference of an electron beam through a polycrystal.
The Experiment
This experiment involves directing a beam of electrons through a carbon target,
scattering the electrons, and analyzing the pattern produced on a luminescent screen.
APPARATUS
THE ELECTRON GUN
The heater, cathode, and anode make up the electron gun. Electrons are
produced by heating a filament that is located inside an oxide-coated metal “can” called
the cathode. Electrons are ejected by thermionic emission from this heated piece of
metal (see Figure 4).
Once emitted, the electrons are accelerated by two pairs of anode rings. You
apply an adjustable acceleration potential Va (2000-4000 V dc) with the kilovolt power
supply between the cathode (-) and anode (+). This accelerates electrons to form an
electron beam.
The beam will have kinetic energy equal to the change in electric potential
energy (eVa ). If the beam velocity is non–relativistic (eVa<<mc2), we have at the anode:
1
2
mv
2
= eV a
(3)
CARBON TARGET
As the electron beam passes into the anode, it strikes a very fine nickel screen
which holds vaporized graphite (carbon). The carbon suspension acts as a polycrystal,
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Electron Diffraction
which means that the carbon consists of a large number of microcrystals that are
randomly oriented. This allows us to sample all possible angles of incidence without
changing the direction of the electron beam. If electrons act like a wave, different atomic
planes will produce constructive interference and the resulting electron diffraction
pattern will consist of concentric rings – one for each plane that satisfies the Bragg’s
Law for constructive interference, as in Figure 3. If electrons act like particles the pattern
will be as shown in Figure 1. Diffraction caused by the nickel mesh may be neglected.
LUMINESCENT SCREEN
After leaving the target, diffracted electrons travel a certain distance (L) and
strike a phosphor screen. The screen is excited by the electrons, allowing you to see the
pattern made by the scattered electrons.
Since the screen is inside a sphere, you should use a flexible ruler to take
measurements off the screen.
Procedure
Use the following diagram to connect the electron diffraction tube to its power supplies.
Screen
6.3 V ac
anode
cathode
F3
F4
R
C5
2 mm plug
heater
G7
carbon target
A
+
Va
ammete
0-5000 V dc
5 kV POWER SUPPLY
Figure 4. Electron Diffraction Experiment Diagram.
CONNECTIONS
Make sure all power is off (all equipment) before making any connections.
Connect the electron diffraction tube to the power supply as shown.
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Electron Diffraction
Connect the multimeter (ammeter) between the anode and the positive side of the high
voltage power supply.
Set the multimeter to its current scale (mA) and turn it on.
Make sure the high voltage supply is set to its minimum. Turn on the power and wait
for one minute for the cathode to reach a stable temperature.
CONTROLLING THE CURRENT
Normal operating current should be kept between 0.05 to 0.10 mA during
measurements, and the voltage should be kept below 4.0 kV. Higher currents can
damage the carbon target. If it becomes necessary to exceed these levels, make your
measurements quickly and decrease the voltage as soon as you obtain your results.
DO NOT EXCEED 0.20 mA current.
Slowly increase the high voltage until you observe a pattern (around 2000 volts).
Monitor the current closely.
Questions
Pre-laboratory assignment
1. Geometric analysis
Using the Figure 5 below, find a geometric relation between D, L and angle φ in
triangle OAB. You may assume that the angle φ is small.
B
carbon target
D/2
φ
A
o
L
2. Express the wavelength λ of the electron as a function of the accelerating potential
Va -- using the de Broglie relation (1) and relation (3).
3. Express the wavelength of the electron as a function of φ -- using Bragg’s relation (2).
Be sure to keep in mind the relation between θ and φ.
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Electron Diffraction
4. Equate the wavelength λ in questions 2 and 3 and solve for D. Show that D is
proportional to 1/ Va .
MEASUREMENTS
First observe whether the pattern of scattered electrons is consistent with particle
behavior or wave behavior.
If you observe any rings on the screen take readings of the accelerating voltage versus
ring diameters for each ring observed. Measure the diameter arc length (diameter bent
around the screen) using a flexible, transparent ruler.
Start at the minimum voltage and record diameters and voltages at 0.25 kV increments
for each ring observed.
V
1st D
2nd
D
3rd
D
At voltages above 4 kV, make each reading quickly and decrease the voltage to zero
as soon as you are finished.
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Electron Diffraction
Analysis
1. Make a plot of 1/ Va versus D for each ring. Does a straight line fit your data? If so,
use the value of the slope to calculate the plane separation d that corresponds to each
ring (use question 4 from the pre-laboratory assignment). The length L is given to be
L=0.14m.
2. If the crystal structure of carbon is cubic, what would be the spacing of the atoms?
Assume a density of 2 g/cm3 and note that 12 grams of carbon contain Avogadro’s
number of atoms. Compare the theoretical value with the value you obtained in
your experiment and comment on its accuracy.
3. In question 1 of the pre-laboratory assignment you are given a relation between L,
D, and f by extrapolating. In reality the screen has a radius of curvature of R=0.066m
which is not equal to L and makes the geometry a bit more complicated. One can
show that the relation between L, R, D and φ is:

D

R sin( )
2R

φ = arctan 
 L − R + Rcos( D ) 

2R 
Use geometric analysis to show this relation.
carbon target
α
φ
R
x
D/2
L-R
L
Check that by setting R equal to L we find the same relation as in question 1 of the
pre laboratory assignment.
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Electron Diffraction
4. Using your measurements of D at different voltages, evaluate the interatomic
spacing d for each ring (make an average evaluation of d for each ring).
V
λ
1st D
2nd
D
3rd
D
5. Does this correction improve the agreement with the theoretical value of d you found
in question 2?
BONUS QUESTIONS:
1. What acceleration potential Va (approximately) should you apply to the electrons to
produce a 1st order (n=1) ring at 20° (use simple geometry for that and keep in mind
the relation between θ and φ).
2. In a diffraction experiment with ordinary light (wavelength of order 5000A), a
grating is made by scratching thin lines on glass. Why doesn’t that work well for
electrons or x-rays?
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Electron Diffraction